Maximum Likelihood Estimation

In this article, we will discuss Maximum Likelihood Estimation (MLE), also referred to as maximum probability estimation. This theoretical point estimation method operates on the principle that when n sets of sample observations are randomly drawn from a population model, the most reasonable parameter estimator should maximize the probability of obtaining these exact observations from the model. This approach differs from least squares estimation, which focuses on finding parameters that provide the best fit to sample data.

Maximum Likelihood Estimation is a widely used method for parameter estimation, particularly in machine learning applications such as logistic regression models. When implementing MLE, practitioners must select an appropriate probability model and determine initial values for model parameters. The core implementation involves constructing a likelihood function L(θ|X) representing the probability of observing data X given parameters θ, then using optimization techniques (like gradient descent or Newton-Raphson method) to find the θ values that maximize this function. In Python, this can be implemented using scipy.optimize.minimize with negative log-likelihood minimization.

Overall, Maximum Likelihood Estimation serves as a powerful estimation technique applicable to various data analysis and modeling tasks across multiple domains. Mastering this method enables practitioners to effectively solve real-world problems by providing statistically optimal parameter estimates with well-understood asymptotic properties.